Lesson 3.1: From Sentences to Symbols

Introduction

In this lesson, we will explore the fascinating world of formal logic, particularly focusing on how we can translate everyday English statements into symbolic notation. By the end of this lesson, you will have a strong understanding of the differences between simple and compound statements, how to use propositional notation, and why symbolizing helps us clarify logical forms.

Learning Objectives:

• Understand simple versus compound statements.
• Learn to translate English statements into propositional notation using letters.
• Discover why symbolizing reveals logical form and removes ambiguity.
• Acknowledge the scope of connectives and the role of brackets.
• Identify common translation traps such as "unless", "only if", and "neither...nor".

Simple versus Compound Statements

A simple statement is a declarative sentence that can be either true or false, but not both. For example:

• "The sky is blue." (This can be true or false.)
• "2 + 2 = 4." (This is true.)

On the other hand, a compound statement combines two or more simple statements using logical connectives. Let's look at some common connectives:

• And ($\land$)
• Or ($\lor$)
• Not (

eg)

• If...then (

ightarrow)

• If and only if ($\leftrightarrow$)

For example:

• "The sky is blue and it is sunny." This is a compound statement combining two simple statements using the connective and ($p \land q$).

Example:

Let:

• $p$: The sky is blue.
• $q$: It is sunny.

Thus, we can represent the statement as: $$p \land q$$

Translating English Statements into Propositional Notation

To translate English statements into symbolic notation, we often assign letters to represent propositions. This enables us to compactly express logical relationships without ambiguity.

Steps for Translation:

1. Identify the main verb or action in the statement.
2. Determine if it is a simple or compound statement.
3. Use letters to symbolize the simple propositions.
4. Apply the correct logical connectives based on the relationships in the statement.

Example:

Consider the statement: "If it rains, then the ground is wet."

• Let:
• $r$: It rains.
• $w$: The ground is wet.
• The logical representation would be: $$r

ightarrow w$$

Why Symbolising Reveals Logical Form

Symbolizing helps us uncover the logical structure within statements. This reduces ambiguity and clarifies the relationships between propositions. With symbols, we can systematically analyze complex arguments without misunderstanding.

The Role of Brackets

Brackets are crucial in logic. They indicate the scope of operations and help us understand which parts of compound statements apply together. For example:

• Without brackets: $p \land q \lor r$ could mean $(p \land q) \lor r$ or $p \land (q \lor r)$.
• With brackets: $p \land (q \lor r)$ clearly shows that $q$ and $r$ are related first before connecting to $p$.

Example:

Given three propositions:

• $a$: It is raining.
• $b$: The park is closed.
• $c$: I will go to the movies.

The statement "The park is closed or I will go to the movies and it is raining" can be symbolized as: $$b \lor (c \land a)$$

Common Translation Traps

Certain phrases in English can lead to confusion when translating into propositional logic. Let's tackle a few common traps:

1. Unless: The phrase "A unless B" can be tricky. It is logically equivalent to "If not B, then A." So, "The party will happen unless it rains" translates to: $$

eg r

ightarrow p$$

1. Only if: The phrase "A only if B" means that if A is true, then B must also be true. The translation would be: $$a

ightarrow b$$

1. Neither...nor: This phrase negates two statements and indicates that both are false. For example, "Neither A nor B" translates to: $$

$eg a \land $

eg b$$

Example:

Suppose we have: "I will not go to the park unless it is sunny". This translates to: $$

eg s

ightarrow

eg p$ where $s$: It is sunny and $p: I will go to the park.

Conclusion

In this lesson, we have learned how to transition from informal English to formal symbolic logic. Understanding the distinction between simple and compound statements, proper use of connectives, and the importance of clear translation is crucial for logical reasoning. As we proceed to more complex forms of logic, having a solid foundation in these concepts will be invaluable.

Study Notes

• A simple statement is true or false; a compound statement combines multiple simple statements.
• Use letters to represent propositions in symbolization.
• Logical connectives (and, or, not, if…then, if and only if) form the backbone of compound statements.
• Brackets clarify the scope of the connections in logical expressions.
• Be wary of common traps like "unless", "only if", and "neither...nor" for accurate translations.